# Difference between revisions of "2005 AIME II Problems/Problem 11"

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Let <math> \displaystyle m </math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of integers such that <math> \displaystyle a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math> \displaystyle m. </math> | Let <math> \displaystyle m </math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of integers such that <math> \displaystyle a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math> \displaystyle m. </math> | ||

− | ''Note: Clearly, the stipulation that the sequence is composed of | + | ''Note: Clearly, the stipulation that the sequence is composed of integers is a minor oversight, as the term <math>\displaystyle a_2 </math>, for example, is obviouly not integral.'' |

== Solution == | == Solution == |

## Revision as of 14:32, 2 March 2007

## Problem

Let be a positive integer, and let be a sequence of integers such that and for Find

*Note: Clearly, the stipulation that the sequence is composed of integers is a minor oversight, as the term , for example, is obviouly not integral.*

## Solution

For , we have

.

Thus the product is a monovariant: it decreases by 3 each time increases by 1. Since for we have , so when , will be zero for the first time, which implies that , our answer.